Theorem coi1in 37941

Description: Precomposition with the identity expressed as intersection. (Contributed by BJ, 16-Aug-2026.)

Assertion

Ref	Expression
coi1in	⊢ (𝐴 ∘ I ) = (𝐴 ∩ (V × V))

Proof of Theorem coi1in

Dummy variables 𝑥 𝑦 𝑧 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		19.42vv 1990	. . . 4 ⊢ (∃𝑦∃𝑧(𝑥 ∈ 𝐴 ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ V ∧ 𝑧 ∈ V))) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ V ∧ 𝑧 ∈ V))))
2		vex 3455	. . . . . . . . . . . . . . 15 ⊢ 𝑡 ∈ V
3	2	ideq 5830	. . . . . . . . . . . . . 14 ⊢ (𝑦 I 𝑡 ↔ 𝑦 = 𝑡)
4	3	anbi1i 636	. . . . . . . . . . . . 13 ⊢ ((𝑦 I 𝑡 ∧ 𝑡𝐴𝑧) ↔ (𝑦 = 𝑡 ∧ 𝑡𝐴𝑧))
5		equcomi 2050	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑡 → 𝑡 = 𝑦)
6	5	breq1d 5113	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑡 → (𝑡𝐴𝑧 ↔ 𝑦𝐴𝑧))
7	6	pm5.32i 585	. . . . . . . . . . . . 13 ⊢ ((𝑦 = 𝑡 ∧ 𝑡𝐴𝑧) ↔ (𝑦 = 𝑡 ∧ 𝑦𝐴𝑧))
8	4, 7	bitri 278	. . . . . . . . . . . 12 ⊢ ((𝑦 I 𝑡 ∧ 𝑡𝐴𝑧) ↔ (𝑦 = 𝑡 ∧ 𝑦𝐴𝑧))
9	8	exbii 1881	. . . . . . . . . . 11 ⊢ (∃𝑡(𝑦 I 𝑡 ∧ 𝑡𝐴𝑧) ↔ ∃𝑡(𝑦 = 𝑡 ∧ 𝑦𝐴𝑧))
10		ax6evr 2048	. . . . . . . . . . . 12 ⊢ ∃𝑡 𝑦 = 𝑡
11		19.41v 1982	. . . . . . . . . . . 12 ⊢ (∃𝑡(𝑦 = 𝑡 ∧ 𝑦𝐴𝑧) ↔ (∃𝑡 𝑦 = 𝑡 ∧ 𝑦𝐴𝑧))
12	10, 11	mpbiran 722	. . . . . . . . . . 11 ⊢ (∃𝑡(𝑦 = 𝑡 ∧ 𝑦𝐴𝑧) ↔ 𝑦𝐴𝑧)
13		df-br 5104	. . . . . . . . . . 11 ⊢ (𝑦𝐴𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐴)
14	9, 12, 13	3bitri 300	. . . . . . . . . 10 ⊢ (∃𝑡(𝑦 I 𝑡 ∧ 𝑡𝐴𝑧) ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐴)
15		eleq1 2849	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 ∈ 𝐴 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐴))
16	14, 15	bitr4id 293	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (∃𝑡(𝑦 I 𝑡 ∧ 𝑡𝐴𝑧) ↔ 𝑥 ∈ 𝐴))
17	16	pm5.32i 585	. . . . . . . 8 ⊢ ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ ∃𝑡(𝑦 I 𝑡 ∧ 𝑡𝐴𝑧)) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑥 ∈ 𝐴))
18	17	biancomi 468	. . . . . . 7 ⊢ ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ ∃𝑡(𝑦 I 𝑡 ∧ 𝑡𝐴𝑧)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝑧⟩))
19		vex 3455	. . . . . . . . 9 ⊢ 𝑦 ∈ V
20		vex 3455	. . . . . . . . 9 ⊢ 𝑧 ∈ V
21	19, 20	pm3.2i 476	. . . . . . . 8 ⊢ (𝑦 ∈ V ∧ 𝑧 ∈ V)
22	21	biantru 539	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝑧⟩) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝑧⟩) ∧ (𝑦 ∈ V ∧ 𝑧 ∈ V)))
23		anass 474	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 = ⟨𝑦, 𝑧⟩) ∧ (𝑦 ∈ V ∧ 𝑧 ∈ V)) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ V ∧ 𝑧 ∈ V))))
24	18, 22, 23	3bitri 300	. . . . . 6 ⊢ ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ ∃𝑡(𝑦 I 𝑡 ∧ 𝑡𝐴𝑧)) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ V ∧ 𝑧 ∈ V))))
25	24	exbii 1881	. . . . 5 ⊢ (∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ ∃𝑡(𝑦 I 𝑡 ∧ 𝑡𝐴𝑧)) ↔ ∃𝑧(𝑥 ∈ 𝐴 ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ V ∧ 𝑧 ∈ V))))
26	25	exbii 1881	. . . 4 ⊢ (∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ ∃𝑡(𝑦 I 𝑡 ∧ 𝑡𝐴𝑧)) ↔ ∃𝑦∃𝑧(𝑥 ∈ 𝐴 ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ V ∧ 𝑧 ∈ V))))
27		elxp 5674	. . . . 5 ⊢ (𝑥 ∈ (V × V) ↔ ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ V ∧ 𝑧 ∈ V)))
28	27	anbi2i 635	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (V × V)) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ V ∧ 𝑧 ∈ V))))
29	1, 26, 28	3bitr4i 306	. . 3 ⊢ (∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ ∃𝑡(𝑦 I 𝑡 ∧ 𝑡𝐴𝑧)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (V × V)))
30		elco 37940	. . 3 ⊢ (𝑥 ∈ (𝐴 ∘ I ) ↔ ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ ∃𝑡(𝑦 I 𝑡 ∧ 𝑡𝐴𝑧)))
31		elin 3915	. . 3 ⊢ (𝑥 ∈ (𝐴 ∩ (V × V)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (V × V)))
32	29, 30, 31	3bitr4i 306	. 2 ⊢ (𝑥 ∈ (𝐴 ∘ I ) ↔ 𝑥 ∈ (𝐴 ∩ (V × V)))
33	32	eqriv 2758	1 ⊢ (𝐴 ∘ I ) = (𝐴 ∩ (V × V))

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  Vcvv 3451   ∩ cin 3898  ⟨cop 4590   class class class wbr 5103   I cid 5545   × cxp 5649   ∘ ccom 5655

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-id 5546  df-xp 5657  df-rel 5658  df-co 5660

This theorem is used by: (None)